Discrete-time AffineQ Term Structure Models with Generalized Market Prices of Risk Anh Le, Kenneth J. Singleton, and Qiang Dai
1
This draft: February 8, 2009
1 Le
is with the University of North Carolina at Chapel Hill, anh
[email protected]. Singleton is with the Graduate School of Business, Stanford University, Stanford, CA 94305 and NBER,
[email protected]. Dai is with Capula Investment Management LLP.
Abstract This paper develops a rich class of discrete-time, nonlinear dynamic term structure models (DT SM s). Under the risk-neutral measure Q, the distribution of the state vector Xt resides within a family of discrete-time affine processes that nests the exact discrete-time counterparts of the entire class of continuous-time models in Duffie and Kan (1996) and Dai and Singleton (2000). Moreover, we allow the market price of risk Λt , linking the risk-neutral and historical distributions of X, to depend generally on the state Xt . The conditional likelihood functions for zero-coupon bond yields for the resulting nonlinear models under the historical measure are known exactly in closed form. As an illustration of our approach, we develop an equilibrium, nonlinear term structure model in which agents exhibit habit formation. Though nonlinear, by design this model shares many of the features of habit-based models in the literature. Moreover, zero-coupon bond prices and the conditional likelihood function of bond yields, consumption growth, and inflation are known in closed form. When evaluated at the maximum likelihood estimates of the parameters, our habit-based model is not able to match key features of the conditional distribution of bond yields.
1
Introduction
This paper develops a rich class of discrete-time, nonlinear dynamic term structure models (DT SM s) in which zero-coupon bond yields and their conditional densities are known exactly in closed form. Under the risk-neutral measure Q, the distribution of the state vector Xt resides within a family of discrete-time affineQ processes1 that nests the exact discrete-time counterparts of the entire class of continuous-time models in Duffie and Kan (1996) and Dai and Singleton (2000).2 Moreover, we allow the market price of risk Λt , linking the Q and historical (P) distributions of X, to depend generally on the state Xt , requiring only that this dependence rules out arbitrage opportunities and that the P distribution of X satisfy certain stationarity/ergodicity conditions needed for econometric analysis. This flexibility in specifying Λt leads to a family of DTSMs in which the conditional P-distributions of Xt+1 and bond yields can show very rich nonlinear dependence on Xt . While this leads immediately to a much richer family of arbitrage-free, affineQ DT SM s than has heretofore been implemented econometrically,3 the primary motivation for this paper derives from the growing literature on equilibrium macro-finance models of the term structure. In particular, the literature on integrating DT SM s with linearized neo-Keynesian (“IS-LM” style) macroeconomic models (e.g., Rudebusch and Wu (2008), Hordahl, Tristani, and Vestin (2007), Wu (2005), and Bekaert, Cho, and Moreno (2006)) has focused exclusively on discrete-time Gaussian DT SM s.4 Arbitrage-free DTSMs are overlaid onto log-linear macro models with Gaussian, homoskedastic shocks. Concurrently, there is a growing literature exploring the ability of preference-based, equilibrium DTSMs to resolve various empirical asset pricing puzzles. Campbell and Cochrane (1999) and Wachter (2005), for instance, develop DTSMs in which agents’ preferences exhibit external habit formation. Alternatively, Bansal and Shaliastovich (2007) and Wu (2008) examine the properties of DTSMs in which agents exhibit preferences for the early resolution of uncertainty and face “long-run” risks in their consumption streams. To date, most of these models have been evaluated using calibrated parameters rather than at estimates from the model-implied likelihood functions. The focus on Gaussian models in the macro-finance literature appears to be driven largely by the absence of tractable discrete-time, multi-factor DT SM s with flexible market prices 1
We use the notation affineQ to denote processes that are affine under the risk neutral measure Q. Our analysis extends immediately to the case of quadratic-Gaussian models discussed in Beaglehole and Tenney (1991), Ahn, Dittmar, and Gallant (2002) and Leippold and Wu (2002). This can be seen from the work of Cheng and Scaillet (2002) who show that quadratic-Gaussian models can be reinterpreted as affine models, after an appropriate expansion of the state vector. 3 With few exceptions, econometric specifications under P of continuous-time, affineQ DTSMs have chosen market prices of risk that preserve the affine structure under P (see, e.g., Dai and Singleton (2000), Duffee (2002), and Cheridito, Filipovic, and Kimmel (2005)). In discrete-time, most of the empirical literature has focused on the even more restrictive case of P and Q Gaussian models. Ang and Piazzesi (2003) and Ang, Dong, and Piazzesi (2007) are examples of studies focusing on monetary policy, while Dai and Philippon (2005) examine fiscal policy within a Gaussian DT SM . 4 Often, agent’s preferences do not appear explicitly in these models, but they are implicit in the specification of the aggregate demand or “IS” function. 2
1
of risk and stochastic volatility. The use of calibration methods rather than likelihood-based estimators in the preference-based literature has been influenced, no doubt in part, by the computational burden associated with the absence of close-form solutions for bond prices. Our proposed framework explicitly addresses both of these issues. Moreover, we overcome many of the challenges with estimation in the literature on continuous-time diffusions. Even when the state vector follows a continuous-time affine diffusion under the physical measure, the one-step ahead conditional density of the state vector is not known in closed form, except for the special cases of Gaussian (Vasicek (1977)) and independent square-root diffusions (Cox, Ingersoll, and Ross (1985)). Accordingly, in estimation, the literature has relied on approximations, with varying degrees of complexity, to the relevant conditional P-densities.5 By shifting to discrete time, we obtain exact representations of the likelihood functions of bond yields even for our most flexible nonlinear models. In particular, we have known likelihood functions for the (discrete-time counterparts to the) entire class of affine DT SM s classified by DS. Therefore, no approximations are necessary in estimation. To illustrate our modeling strategy we develop a habit-based model of the term structure of interest rates, starting from the pricing kernel examined in Campbell and Cochrane (1999) (hereafter CC), Wachter (2005), and Verdelhan (2008). These authors posit affine representations of the state under P which, when combined with the habit-based pricing kernel, lead to nonlinear expressions for bond prices that must be solved numerically. Moreover, likelihood functions for the data are not known in closed form. Instead, we assume that Xt follows an affineQ process of a form that embeds all of the key features of extant models with habit formation, including time-varying volatility of the surplus consumption ratio St , nonzero correlation between this ratio and inflation πt , and an implied persistence in consumption growth. The market prices of risk associated with our habit-based DT SM and state process give rise to a nonlinear (non-affine) representation of bond yields under the historical distribution. Nevertheless, we show that, by appropriate choice of the consumption growth process, an equilibrium implication of our model is that the short rate is an affine function of the state. Consequently, zero-coupon bond yields are affine functions of the state. Moreover, the likelihood function of the data is known in closed form. CC, Wachter, and Verdelhan calibrated their model to selected sets of parameters. Others have used GM M methods to estimate equilibrium models off Euler equations; see, for example, Fuhrer (2000) and Engsted and Moller (2008) for habit-based models, and Bansal, Kiku, and Yaron (2007) and Constantinides and Ghosh (2008) for models with long-run risks in consumption growth. Our framework renders full-information maximum likelihood feasible for these (and other) equilibrium asset pricing models. We proceed to compute ML estimates of our habit-based model using historical data on consumption growth, inflation, and U.S. Treasury bond yields. We compare our estimates, 5
These include the direct approximations to the conditional densities explored in Duan and Simonato (1999), Ait-Sahalia (1999, 2002), and Duffie, Pedersen, and Singleton (2003); the Monte Carlo based approximations of Pedersen (1995) and Brandt and Santa-Clara (2001)); and the simulation-based method-ofmoments estimators proposed by Duffie and Singleton (1993) and Gallant and Tauchen (1996).
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and the model-implied properties of the conditional distribution of bond yields, to those implied by parameters chosen according to several sensible calibration schemes. The results highlight some of the limitations of the habit-based models that have been examined to date. In what is perhaps the closest precursor to our construction of arbitrage-free pricing models, Gourieroux, Monfort, and Polimenis (2002) developed DT SM s based on the single-factor autoregressive gamma model (the discrete-time counterpart to a one-factor CIR model), and multi-factor Gaussian models (the counterparts of AQ 0 (N ) models). In terms of coverage of models, our framework extends their analysis to all of the families of multi-factor models DAQ M (N ), M = 0, 1, . . . , N . Furthermore, Gourieroux, et. al. assumed that the market price of risk Λ is constant and, as such, they focused on the “completely” affine versions Q of the DAQ 1 (1) and DA0 (N ) models. A major focus of our analysis is on the specification and estimation of discrete-time affine DTSMs that allow general dependence of Λt on Xt . Moreover, we illustrate this flexibility by computing ML estimates of an equilibrium asset pricing model using both macroeconomic and bond market data. The remainder of this paper is organized as follows. We start in Section 2 with a more in depth motivation for our modeling framework using the habit-based asset pricing model introduced by Campbell and Cochrane (1999). We then proceed to develop both the theoretical properties of our modeling approach and their application to a habit-based DT SM in parallel. Section 3 presents the canonical families of affineQ processes DAQ M (N ), 0 ≤ M ≤ N . The specific formulations of the state process in the habit-based model are set forth in Section 4, and closed-form expressions for equilibrium bond prices are derived. In the process, we also demonstrate that, as an equilibrium implication of our formulation, the short rate is an affine function of the surplus consumption ratio and inflation rate. The distribution of bond yields under the physical measure is taken up in Section 5. For D each family DAQ M (N ), we specify an associated family of state-price densities (dP/dQ)t+1 linking the P and Q distributions of Xt+1 that has a natural interpretation as a discrete-time counterpart to the state-price density associated with affine diffusion-based, continuous-time DTSMs. Moreover, just as in a continuous-time model, we allow the modeler substantial flexibility in specifying the dependence of the market price of factor risks, Λt , on Xt . By roaming over admissible choices of Λt , we are effectively ranging across the entire family of admissible arbitrage-free DTSMs constructed under the assumption that, under Q, X follows a process residing in one of the families DAQ M (N ). Importantly, a key difference between our discrete-time construction and the continuous-time counterpart is that each choice of Q (dP/dQ)D t+1 , when combined with a known affine distribution of the state X, leads to a known parametric representation of the P-distribution of bond yields. The properties of the market prices of risk underlying our choice of (dP/dQ)D t+1 are elaborated on in Section 6. Details of the P distribution of the state and the associated market prices of risk in our habit-based, illustrative DT SM are presented in Section 7. Finally, the empirical examples are presented in Section 8.
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2
An Illustrative Model with Habit Formation
Following CC and Wachter (2005), we assume that agents maximize the utility function: E
∞ X t=0
δt
(Ct − Ht )1−γ − 1 , 1−γ
(1)
where Ht is the level of habit, δ is the subjective discount factor and γ is the utility curvature parameter. The consumption surplus ratio is defined as St = (Ct − Ht )/Ct . For any asset with nominal (total) return Rt+1 , this leads to the Euler equation " µ # ¶−γ µ ¶−γ Ct+1 Pt St+1 Et δ Rt+1 = 1, (2) St Ct Pt+1 where Pt is the price level in this economy at date t. The nominal pricing kernel, in natural logarithm, can be written as: mt,t+1 = log δ − γ(st+1 − st ) − γgt+1 − πt+1
(3)
where st = log(St ), gt+1 = log (Ct+1 /Ct ) and πt+1 = log (Pt+1 /Pt ). As in CC and Wachter (2005), we assume that the state vector Xt is comprised of the current consumption surplus ratio, st , and current inflation rate, πt . Consumption growth, gt , is assumed to be conditionally perfectly correlated with the consumption surplus ratio. Finally, the upper bound of st is captured by a free parameter, smax .6 We let zt = smax − st denote the inverse consumption surplus ratio. Several practical issues arise when flushing out an implementable version of this model with habit formation. First, z is a strictly positive process and, therefore, innovations in consumption growth (equivalently, zt ) cannot literally be Gaussian as assumed by Wachter. We circumvent this inconsistency by directly positing a strictly positive, discrete-time stochastic process for the inverse consumption ratio. As in CC and Wachter, our representation of zt also exhibits conditional heteroskedasticity. Additionally, in illustrative models of habit formation, researchers have typically assumed that consumption growth follows an affine process under P; CC and Wachter, for example, assume that gt is an i.i.d. process (consumption follows a random walk with drift). This, together with their model for st , implies that bond prices must be determined numerically from the representative agent’s Euler equation. Instead, we formulate our model so that Xt follows an affineQ process and the one-period short-rate rt is an affine function of Xt , and this leads immediately to closed-form solutions for zero-coupon bond prices (Duffie and Kan (1996)). Then market prices of risk are chosen so that the P distribution of consumption growth shares many of the features of previous specifications. In fact, in the continuous-time 6
Note that st is always negative therefore a natural (trivial) upper bound for st is smax = 0. st = 0 implies a zero habit level: Ht = 0. Therefore a non-zero upper bound of st essentially imposes a minimum level of habit, Ht , as a fraction of current consumption, Ct .
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limit of our discrete-time model, gt is i.i.d. and conditionally homoskedastic, just as in CC and Wachter. In this equilibrium setting, the functional dependence of rt on Xt depends on the structure of preferences and the specifications of the P and Q distributions of the state. As part of the development of our pricing model with habit formation, we demonstrate that the affine dependence of rt on Xt is an equilibrium implication of the model. This is achieved by judicious choice of the drift of gt under Q. We expand on these and related issues subsequent to presenting the affineQ family of models used in our empirical illustrations.
Canonical Discrete-Time AffineQ Processes
3
Following Duffie, Filipovic, and Schachermayer (2003), we will refer to a Markov process X as affineQ if the conditional Laplace transforms of Xt+1 given Xt is an exponential-affine function of Xt :7 under a probability measure Q, for an N × 1 state vector X, ¯ i h 0 u Xt+1 ¯ Q Q φ (u; Xt ) = E e (4) ¯ Xt = ea(u)+b(u)Xt . Paralleling DS, we focus (by choice of the N × 1 vector a(u) and N × N matrix b(u)) on the particular sub-families of discrete-time affine models DAQ M (N ) that are formally the 8 exact discrete-time counterparts to their families AQ (N ). The members of DAQ M M (N ) are well-defined affine models in their own right, and also have (by construction) the property that, as the sampling interval of the data shrinks to zero, they converge to members of the continuous-time family AQ M (N ). Throughout this paper, we assume that the state vector Xt is affine under the risk-neutral measure Q, in the sense just described. Hence equation (4) constitutes a basic distributional assumption of our model. In the rest of this section, we make explicit the functional forms of a(·) and b(·) that define the Q-affine families DAQ M (N ), M = 0, . . . , N .
3.1
DAQ 0 (N )
The DAQ 0 (N ) process is an N × 1 vector Y that follows a Gaussian vector autoregression: conditional on Yt , Yt+1 is normally distributed with conditional mean µ0 + µY Yt , and conditional covariance matrix V . The conditional Laplace transform of Y is given by (4) with 1 a(u) = µ00 u + u0 V u, b(u) = u0 µY . 2 7
(5)
See Duffie, Pan, and Singleton (2000) for a proof that the continuous-time affine processes typically examined have conditional characteristic functions that are exponential-affine functions, and Gourieroux and Jasiak (2006) and Darolles, Gourieroux, and Jasiak (2006) for discussions of discrete-time affine processes related to those examined in this paper. 8 These are not the only well-defined discrete-time affine DTSMs. Gourieroux, Monfort, and Polimenis (2002) discuss a variety of other examples that are outside the purview of our analysis (because their continuous-time counterparts do not reside in one of the families AQ M (N )).
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To derive the continuous-time counterpart of this family, let ∆t be the length of the observation interval, and let µ0 = κQ θQ ∆t, µY = IN ×N − κQ ∆t, and V = σσ 0 ∆t, where κQ and σ are N × N matrices and θQ is a N × 1 vector. Then in the limit ∆t → 0, the process DAQ 0 (N ) converges to the continuous-time A0 (N ) process, the N -dimensional Gaussian process: dYt = κQ (θQ − Yt )dt + σdBtQ , where BtQ is a N × 1 vector of standard Brownian motions under the measure Q.
3.2
DAQ N (N )
The DAQ N (N ) process is the exact discrete-time equivalent of the multi-variate correlated square-root or CIR process; Z is non-negative with probability one, no approximations are required in the pricing of bonds, and the associated likelihood functions are known exactly in closed-form. The scalar case N = 1 was explored in depth in Gourieroux and Jasiak (2006) and Darolles, Gourieroux, and Jasiak (2006). We extend their analysis to the multi-variate case of a DAQ N (N ) process Zt as follows. As in the canonical AQ N (N ) model of DS we assume that, conditional on Zt , the components of Zt+1 are independent. To specify the conditional distribution of Zt+1 , we let % be an N × N matrix with elements satisfying 0 < %ii < 1, %ij ≤ 0, 1 ≤ i, j ≤ N. Furthermore, for each 1 ≤ i ≤ N , we let ρi be the ith row of the N × N non-singular matrix ρ = (IN ×N − %). Then, for constants ci > 0, νi > 0, i = 1, . . . , N, we define the conditional i density of Zt+1 given Zt as the Poisson mixture of standard gamma distributions: i Zt+1 |(P, Zt ) ∼ gamma(νi + P), where P|Zt ∼ P oisson(ρi Zt /ci ). (6) ci Here, the random variable P ∈ (0, 1, 2, . . .) is drawn from a Poisson distribution with intensity modulated by the current realization of the state vector Zt , and it in turn determines the coefficient of the standard gamma distribution (with scale parameter equal to 1) from which i Zt+1 is drawn. i The conditional density function of Zt+1 takes the form: ³ i ´k ³ i ´νi +k−1 Zt+1 Z − ρ Z t+1 i t ∞ e ci ρ Z ci ci 1 X − ic t Q i . i f (Zt+1 |Zt ) = e × (7) ci k=0 k! Γ(νi + k)
Using conditional independence, the distribution of a DAQ N (N ) process Zt+1 , conditional on QN Q i Q Zt , is given by f (Zt+1 |Zt ) = i=1 f (Zt+1 |Zt ). Finally, it is straight-forward to show that for any u, such that ui < c1i , the conditional Laplace transform of Zt+1 is given by (4) with a(u) = −
N X
νi log (1 − ui ci ), b(u) =
i=1
N X i=1
6
ui ρi . 1 − ui c i
(8)
When the off-diagonal elements of the N ×N matrix % are non-zero, the autoregressive gamma processes {Z i } are (unconditionally) correlated. Thus, even in the case of correlated Zti , the conditional density of Zt+1 is known in closed form. This is not the case for correlated Z in i j the continuous-time family AQ N (N ). The nature of the correlation between Z and Z (i 6= j) is constrained by our requirement that %ij ≤ 0. Analogous to the constraint imposed by DS on the off-diagonal elements of the feedback matrix κQ in their continuous-time models, this constraint serves to ensure that feedback among the Z’s through their conditional means does not compromise the requirement that the intensity of the Poisson process be positive. Equivalently, it ensures that we have a well-defined multivariate discrete-time process taking on strictly positive values. The conditional mean EtQ [Zt+1 ] and conditional covariance matrix VtQ [Zt+1 ] implied by the conditional moment-generating function (4) and (8) are EtQ [Zt+1 ](i) = νi ci + ρi Zt , VtQ [Zt+1 ](i, i) = νi c2i + 2ci ρi Zt ,
(9)
and the off-diagonal elements of VtQ [Zt+1 ] are all zero (correlation occurs only through the feedback matrix). Note the similarity between the affine form of these moments and those of the exact discrete-time process implied by a univariate square-root diffusion. 9 That this process converges to the multi-factor correlated AQ N (N ) process can be seen 2 Q Q σ by letting ρ = IN ×N − κQ ∆t, ci = 2i ∆t, and νi = 2(κσ2θ )i , where κQ is a N × N matrix and i
θQ is a N × 1 vector. In the limit as ∆t → 0, the DAQ N (N ) process converges to: p dZt = κQ (θQ − Zt )dt + σ diag(Zt )dBtQ , where σ is a N × N diagonal matrix with ith diagonal element given by σi . 3.2.1
DAQ M (N ) Processes, For 0 < M < N
We refer to an N × 1 vector of stochastic processes Xt = (Zt0 , Yt0 )0 as a DAQ M (N ) process if Q (i) Zt is an autonomous DAM (M ) process; and (ii) the Laplace transform of f Q (Xt+1 |Xt ) = f Q (Yt+1 |Zt+1 , Yt , Zt ) × f Q (Zt+1 |Zt ),
(10)
is an exponential-affine function of Xt . This will be the case if Yt+1 is exponentially affine with respect to (Zt+1 , Yt , Zt ).10 For example, if f Q (Yt+1 |Zt+1 , Yt , Zt ) is the density of a Gaussian process with conditional mean and variance ωYQt ≡ µ0 + µZ Zt+1 + µY Xt and ΩY t ≡ ΣY SY t Σ0Y , where ΣY is an (N − M ) × (N − M ) matrix, and SY t is a (N − M ) × (N − M ) diagonal matrix with the ith diagonal Gourieroux and Jasiak (2006) attribute the insight that the DAQ 1 (1) process is a discrete-time counterpart to the square-root diffusion to Lamberton and Lapeyre (1992). 10 To see this, consider: E[euY Yt+1 +uZ Zt+1 |Yt , Zt ] = E[E[euY Yt+1 |Zt+1 , Yt , Zt ]euZ Zt+1 |Yt , Zt ]. If Yt+1 is exponentially affine with respect to (Zt+1 , Yt , Zt ) then E[euY Yt+1 |Zt+1 , Yt , Zt ] = eaZ Zt+1 +bY Yt +cZ Zt which implies E[euY Yt+1 +uZ Zt+1 |Yt , Zt ] = E[e(aZ +uZ )Zt+1 |Zt ]ebY Yt +cZ Zt which is exponential-affine in Xt . 9
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element given by αi + βi0 Zt , 1 ≤ i ≤ N − M . 11 The proposed formulation of a habit-based DT SM follows this structure. We will assume that the inverse surplus consumption ratio zt+1 follows a DAQ 1 (1) process and inflation πt+1 is Gaussian conditional on (zt+1 , Xt ), and this will be shown to imply that Xt follows an affineQ process.
3.3
Bond Pricing
As in the extant literature on affine term structure models, we assume that the interest rate on one-period zero-coupon bonds is an affine function of the state: rt = δ0 + δX Xt , where δX > 0 is a 1 × N vector.12 With this additional assumption, the time-t zero-coupon bond price with maturity of n periods is given by h Pn−1 i £ n−1 ¤ Dtn = EtQ e− i=0 rt+i = e−rt EtQ Dt+1 = e−An −Bn Xt , (11) where the loadings An and Bn are determined by the following recursion: An − An−1 = δ0 + An−1 − a(−Bn−1 ), Bn = δX − b(−Bn−1 ),
(12) (13)
with the initial condition A0 = B0 = 0.13
4
Pricing in the Habit-Based DT SM
In this section we apply the framework just presented to the pricing of nominal zero-coupon bonds in the habit-based DT SM . We proceed in three steps: first we present the riskneutral, affineQ representation of the state; then we show that, by appropriate choice of the drift of consumption growth, in equilibrium the short-rate rt is affine in Xt ; and finally we combine these results to drive close-form expressions for zero-coupon bond prices.
4.1
Risk-Neutral Representation of the State
The inverse consumption surplus ratio: 11
For continuous-time formulations, Collin-Dufresne, Goldstein, and Jones (2008) and Joslin (2007) show that, when N ≥ 4 and 2 ≤ M ≤ N − 2, then this formulation of the conditional variance is not the maximal canonical AQ M (N ) model. Our framework accommodates the discrete-time counterpart to their maximal models by appropriate choice of ΩY t . 12 If Xt is a DAQ M (N ) process, then setting δXi > 0 for i > M is a normalization, but setting δXi > 0 for i ≤ M is a model restriction. When M > 0, this restriction ensures that (i) the level of the short rate r and the factors with stochastic volatility are positively correlated; and (ii) zero-coupon bond prices are well defined for any maturity. See Footnote 13 for further elaboration on the second point. 13 When M > 0, the assumption δX > 0 ensures that the first M elements of Bn are never negative. This in turn ensures that a(·) and b(·) are always evaluated in their admissible range in the recursion.
8
Since the inverse surplus consumption ratio zt is strictly positive, it is natural to model zt as a DAQ 1 (1) process. That is: µ ¶ zt+1 ρ z zt |(P, zt ) ∼ gamma (νz + P) , and P|zt ∼ P oisson . (14) cz cz The first two conditional moments of zt are: EtQ [zt+1 ] = ρz zt + vz cz σtQ [zt+1 ]2 = 2ρz cz zt + vz c2z
(15) (16)
The consumption growth: We assume that under Q consumption growth follows the process gt+1 = f (zt ) − σg
zt+1 − EtQ [zt+1 ] . σtQ [zt+1 ]
(17)
The innovation in gt+1 , zt+1 −EtQ [zt+1 ], is the shock to st+1 ; that is, st+1 and zt+1 are perfectly correlated conditional on date t information. The scaling by σtQ [zt+1 ] renders gt+1 approximately conditionally homoskedastic, an assumption maintained in both CC and Wachter.14 The conditional mean of consumption growth, f (zt ), will be chosen subsequently to ensure that, in equilibrium, the short rate rt is an affine function of the state. The inflation process: Following Wachter we assume that inflation has no impact on the real side of the economy. However, we do allow both for nonzero correlation between the innovations in πt+1 and zt+1 , as well as feedback from the real side of the economy to inflation (zt affects πt+1 ): πt+1 = π ¯ + ρπ (πt − π ¯ ) + ρπ,z (zt − E Q [zt ]) − σπ,g (zt+1 − EtQ [zt+1 ]) + σπ ²Q π,t+1 ,
(18)
where ²Q π,t+1 ∼ N (0, 1) and the risk-neutral long run Q-mean of zt is νz cz /(1 − ρz ). The parameters ρπ and σπ govern the autoregressive nature of inflation and idiosyncratic inflation shocks, respectively. The parameters ρπ,z and σπ,g modulate the unconditional and conditional correlation between consumption growth and inflation. Risk-neutral density of states: In the notation of the last section, Xt follows a DAQ 1 (2) with one-period ahead density f Q (zt+1 , πt+1 |zt , πt ) = f Q (zt+1 |zt ) × f Q (πt+1 |zt+1 , zt , πt )
(19)
where f Q (zt+1 |zt ) is given by equation (7) and f Q (πt+1 |zt+1 , zt , πt ) is a Gaussian density with ¯ + ρπ (πt − π ¯ ) + ρπ,z (zt − E Q [zt ]) − σπ,g (zt+1 − EtQ [zt+1 ]) (20) EtQ [πt+1 |zt+1 , zt , πt ] = π (21) σtQ [πt+1 |zt+1 , zt , πt ] = σπ . gt+1 is exactly conditionally homoskedastic if σtQ [st+1 ] = σtP [st+1 ]. However, it can be shown that the difference between these two quantities is small for typical sampling intervals ∆, on the order of (∆)2 . 14
9
Given this structure, it follows immediately that X is an affineQ process with Laplace transform φQ (u; [zt , πt ]) = EtQ [euz zt+1 +uπ πt+1 ] = ea(u)+bz (u)zt +bπ (u)πt , (22) where:
µ
a(u) = uπ
ν z cz π ¯ (1 − ρπ ) − ρπ,z + σπ,g νz cz 1 − ρz
¶
1 + σπ2 u2π − νz log (1 − (uz − uπ σπ,g )cz ) (23) 2
and bz (u) = uπ (ρπ,z + σπ,g ρz ) +
ρz (uz − uπ σπ,g ) 1 − (uz − uπ σπ,g )cz
bπ (u) = uπ ρπ .
4.2
(24) (25)
Bond Prices in the Habit-Based DT SM
Key to obtaining closed-form representations of bond prices are the conditions that Xt follows an affineQ process and rt is an affine function of Xt . The former property of the model is introduced by assumption on the exogenous variables in the model. We turn next to a sufficient set of restrictions on the risk-neutral expectation of consumption growth to ensure that the model-implied, equilibrium short rate is an affine function of Xt . Proposition 1 If the conditional expectation of gt+1 under Q is given by: Ã ! Q uΛ zt+1 1 E [e ] t f (zt ) = C − (γ + σπ,g )σg σtQ [zt+1 ] − log QG uΛ zt+1 γ Et [e ]
(26)
where • C is a constant ³ • uΛ = −γ 1 + σQ [zσg t
t+1 ]
´ − σπ,g
• QG denotes a Gaussian measure with the same conditional mean and variance implied by the measure Q, then the nominal interest rate per unit of time interval is affine in the state: r t = δ 0 + δ z z t + δ π πt
(27)
where δπ = ρπ ,15 νz cz − γνz cz + γC 1 − ρz 1 1 1 + γ 2 σg2 + (γ + σπ,g )2 νz c2z + σπ2 , 2 2 2 2 = γ(1 − ρz ) + ρπ,z + (γ + σπ,z ) ρz cz .
δ0 = −log δ + (1 − ρπ )¯ π − ρπ,z
δz 15
(28) (29)
At first glance, the fact that rt increases in σg2 and σπ2 might seem contrary to investors’ pre-cautionary savings motive. However, this is a consequence of representing δ0 and δz in terms of parameters of the risk-neutral distribution. If the risk-neutral mean is replaced by its equivalent expression in terms of the physical mean and market prices of risk, then we recover the usual negative coefficients on volatilities.
10
Proof: See Appendix D. We defer further interpretation of the nonlinear conditional Q-mean f (zt ) of consumption growth until after we have specified the market prices of risk. This will allow direct comparisons between the model-implied P and Q distributions of surplus consumption and consumption growth. From Proposition 1 and our assumption that the states follow a DAQ 1 (2) process, it follows that nominal zero-coupon bond prices of any maturity are exponentially affine in the state.
5
Physical Distribution of Bond Yields
A standard means of constructing an affine DTSM in continuous time is to start with a representation of X in one of the families AQ M (N ) and then to specify a market price of risk ηt that defines the change of measure from Q to P for X. In principle, starting with an affineQ model for X, one can generate essentially any functional form for the P drift of X by choice of the market price of risk η, up to the weak requirement that η not admit arbitrage opportunities. What has led researchers to focus on relatively restrictive specifications of η(Xt ) are the computational burdens of estimation that arise when the chosen η leads to an unknown (in closed form) P-likelihood function for the observed bond yields. In this section we introduce a discrete-time P-formulation of affine DTSMs that overcomes this limitation of continuous-time models. This is accomplished by choosing a RadonNykodym derivative (dP/dQ)D (Xt+1 , Λt ) satisfying f P (Xt+1 |Xt ) = (dP/dQ)D (Xt+1 ; Λt ) × f Q (Xt+1 |Xt ),
(30)
with the properties that (P1) it is known in closed form (so that f P can be derived in closedform from our knowledge of f Q developed in Section 3); (P2) Λt is naturally interpreted as the market price of risk of Xt+1 ; and (P3) rich nonlinear dependence of Λt on Xt is accommodated. In principle, any choice of (dP/dQ)D that is a known function of (Xt+1 , Λt ) and for which P and Q are equivalent measures (as required by the absence of arbitrage) leads to a nonlinear DTSM satisfying P1. We proceed by adopting the following particularly tractable choice of (dP/dQ)D : µ ¶D 0 dP eΛt Xt+1 (Xt+1 ; Λt ) = Q , (31) dQ φ (Λt ; Xt ) where φQ is the conditional Laplace transform of X under Q, Λt is a N × 1 vector of functions of Xt satisfying P rob{Λit ci < 1} = 1, for 1 ≤ ∀i ≤ M , and P rob{Λit < ∞} = 1, for M + 1 ≤ i ≤ N . This formulation of (dP/dQ)D is a conditional version of the Esscher (1932) transform for the conditional Q distribution of X.16 With this choice of (dP/dQ)D , 16
Buhlmann, Delbaen, Embrechts, and Shiryaev (1996) formally develop the conditional Essher transform using martingale theory in the context of no-arbitrage pricing. A notable application of the Esscher transform (with constant Λ) to option pricing is Gerber and Shiu (1994) who demonstrate that many variants of the Black-Scholes option pricing model can be developed using the Esscher transform. For our purposes, the conditional transform is essential, because of our linkage (see below) of Λt to the market prices of risk.
11
the conditional P-Laplace transform of Xt is given by φP (u; Xt ) =
φQ (u + Λt ; Xt ) = eA(u;Λt )+B(u;Λt )Xt , φQ (Λt ; Xt )
(32)
where A(u; v) ≡ a(u + v) − a(v) and B(u; v) ≡ b(u + v) − b(v). Though φP (u; Xt ) has an exponential-affine form, A(u; Λt ) and B(u; Λt ) are functions of Λt which, in turn, may be a nonlinear function of Xt . Thus, in general X is not an affine process under P. We elaborate on the nature of the non-affine nature of this distribution below. With this choice of (dP/dQ)D , the pricing kernel for pricing one-period ahead payoffs in our discrete-time model is 0
Mt,t+1 ≡ e
−rt
f Q (Xt+1 |Xt ) e−Λt Xt+1 × P = e−rt × P , f (Xt+1 |Xt ) φ (−Λt ; Xt )
(33)
£ ¤−1 where we have used the fact that φP (−Λt ; Xt ) = φQ (Λt ; Xt ) , which follows from (32) evaluated at u = −Λt . This choice of Radon-Nykodym derivative– equivalently pricing kernel M– is natural in that, for small time interval ∆, its counterpart in affineQ diffusion models (dP/dQ)C is approximately equal to (dP/dQ)D (t, t + ∆).17 As such, the P distributions of the bond yields implied by our families DAQ M (N ), and associated market prices of risk Λ, capture essentially the same degree of flexibility inherent in the families AQ M (N ) as one ranges across all admissible (arbitrage-free) specifications of the market prices of risk η(Xt ). It is in this sense that we view our framework as the discrete-time counterpart of the entire family of arbitrage-free, continuous-time affine DTSMs derived under the assumption that the Q-representation of X resides in one of the families AQ M (N ). The restrictions that the products Λit ci , 1 ≤ i ≤ M , for the M volatility factors are bounded by unity are required to ensure that f P is a well-defined probability density function and that P and Q are equivalent measures. This follows from the observation that φQ (u; Xt ) is finite if and only if ui ci < 1. Unless Λit ci < 1 almost surely, for i = 1, . . . , M , φQ (Λt ; Xt ) is infinite with positive probability. In this case, f P would not integrate to unity for a set of Xt that has positive measure, and P and Q would not be equivalent. Examining these restrictions more closely, and using our mapping to the parameters of the related CIR process, we see that we are effectively requiring that 2/(σi2 ∆t) > Λit , i = 1, . . . , M . Typically σi2 is small and, depending on the application, ∆t may also be small. Therefore, 17
P √ThatP is, for a small time interval ∆, and approximate affine state process Xt+∆ ≈ µX (Xt )∆ + ΣX SXt ²t+∆ , with ²t+∆ |Xt ∼ N (0, ∆I), 0
1
(dQ/dP)C t,t+∆
0 P
0
√
P
≈
e− 2 ηt ηt ∆−ηt ²t+∆ e−Λt ΣX SXt ²t+∆ i h 1 0 i h = √ P 0 P 0 EtP e− 2 ηt ηt ∆−ηt ²t+∆ EtP e−Λt ΣX SXt ²t+∆
=
e−Λt Xt+∆ e−Λt Xt+∆ ¤ £ = , 0 φP (−Λt ; Xt ) EtP e−Λt Xt+∆
0
0
¡ √ ¢0−1 where Λt ≡ ΣX SXt ηt is a transformation of the market price of risk ηt .
12
these bounds are typically weak and in the applications we have encountered so far they are far from binding. As ∆t approaches zero (continuous time), the only requirement is that the Λit be finite almost surely. Under these regularity conditions we have all of the information necessary to construct the likelihood function of the state, and hence the bond yields, under P. We effectively know f Q (Xt+1 |Xt ) from the cross-sectional behavior of bond yields.18 Furthermore, the relationship between the observed yields yt and the state vector Xt are also known due to the pricing equation (11), which depends only on the risk-neutral distribution f Q (Xt+1 |Xt ). Thus, the unknown function (dP/dQ)D (Xt+1 ; Λt ) can be estimated from the time-series observations of bond yields, yt .
6
The Market Prices of Risk
An immediate implication of (31) is that, if Λt = 0, then f P (Xt+1 |Xt ) = f Q (Xt+1 |Xt ). Thus, agents’ market prices of risk are zero if and only if Λt = 0. In our discrete-time setting, Λt is not literally the market price of X risk (MPR), but rather the MPR is a nonlinear (deterministic) function of Λt . However, in a sense that we now make precise, Λt is the dominant term in the MPR. Accordingly, we will refer to Λt as the MPR as this will facilitate comparisons with the MPR in continuous-time (AQ M (N ), η) models. Notice first of all that19 £ ¤ £ ¤ EtP [Xt+1 ] − EtQ [Xt+1 ] = A(1) (0; Λt ) − a(1) (0) + B (1) (0; Λt ) − b(1) (0) Xt = VtP [Xt+1 ] × Λt + o(Λt ),
(34)
where VtP [·] is the conditional covariance matrix under P. Ignoring the higher order terms, the above relationship is exactly what arises in diffusion-based models: Λt is the vector of market prices of risk underlying the adjustment to the “drift” in the change of measure from Q to P. Moreover, the continuously compounded, expected excess return on the security 0 with the payoff e−c Xt+1 is · ¸ 0 £ ¤ £ ¤ e−c Xt+1 P Et log Q − rt = − a(−c) + c0 a(1) (Λt ) − b(−c) + c0 b(1) (Λt ) Xt , 0 Et [e−rt e−c Xt+1 ] = −c0 VtP [Xt+1 ] × Λt + o(c) + o(Λt ). (35) Since c determines the exposure of this security to the factor risk X and VtP [Xt+1 ] measures the size of the risk, the random variable Λt is the dominant term in the true market price of risk underlying expected excess returns. 18
Intuitively, taking the leading principal components as the state vector, we can estimate δ0 , δX , An , and Bn by regressing bond yields on this state vector. The parameters that characterize f Q (Xt+1 |Xt ) can then be estimated by treating the recursions (12) and (13) as (possibly nonlinear) cross-equation restrictions. 19 The terms A(1) (0; Λt ) and B(1) (0; Λt ) are the first derivatives of A and B with respect to their first arguments, and a(1) (u) and b(1) (u) are the first derivatives of a(u) and b(u).
13
A notable difference between Λt and the market price of risk ηt that appears in continuoustime (AQ M (N ), η) models is that Λt measures the price of risk per per unit of variance, whereas η measures risk in units of standard deviation. From the heuristic mapping between of our choice of (dP/dQ)D and its continuous-time counterpart (see footnote 17) it is seen that this difference is simply a consequence of our (implicit) convention that ³ p ´0−1 Λt = ΣX SXt ηt .
(36)
Researchers who want to replicate features of a continuous-time affine DT SM , can do so by setting Λt as in (36). For instance, choosing ηt as in Duffee (2002), Duarte (2004), or Cheridito, Filipovic, and Kimmel (2005) would lead to a discrete-time DT SM that locally (for small time interval ∆) would replicate the P moments of their models. More generally, it is evident from (34) that, starting from an affine EtQ [Xt+1 ], essentially any functional form for EtP [Xt+1 ] is achievable by an appropriate choice of Λt . In particular, if one sets Λt ≡ (ΣX SX (t)Σ0X )−1 (µP (Xt ) − µQ (Xt )), (37) p P where Σ S(t) is the diffusion term in an AQ M (N ) affine diffusion model and µ (Xt ) is the desired P-drift of a diffusion model for X, then locally one would obtain E P [Xt+∆ |Xt ] = Xt + µP (Xt )∆ + o(∆) Cov P [Xt+∆ |Xt ] = ΣS(t)Σ0 ∆ + o(∆).
(38) (39)
That is, starting with an affine specification of the Q drift µQ (Xt ), we can generate essentially any desired nonlinear Xt dependence of the P drift of X, µP (Xt ), by choosing Λt as in (37). Of course with Λt set to induce a nonlinear EtP [Xt+1 ], the conditional Esscher transform (31) in general induces nonlinear conditional P moments of all orders, not just a nonlinear conditional mean. For example, letting ΛZt and ΛY t form a conformal partition of Λt , the conditional P-mean of the ith member of the M -vector of volatility factors Zt+1 is ¯ ¯ £ i ¤ ∂ ρi νi ci P Et Zt+1 = [A(u; Λt ) + B(u; Λt )Xt ]¯¯ = + Zt . (40) ∂uZi 1 − ΛZt,i ci (1 − ΛZt,i ci )2 u=0 i Similarly, the conditional variance of Zt+1 is given by i VarPt [Zt+1 ]=
2ci ρi Zt νi c2i + , i = 1, . . . , M. 2 (1 − ΛZt,i ci ) (1 − ΛZt,i ci )3
(41)
The nonlinearity of these moments, in contrast to their affine counterparts under Q (see (9)), is induced by the state-dependence of ΛZt,i through the terms 1/(1 − ΛZt,i ci ). What our formulation of the (DAQ M (N ), Λ) model does not allow is complete freedom in specifying the nonlinearity of higher order moments, once we have chosen a functional form for the conditional first moment. This is illustrated by (40) and (41) where each i ] is divided by one higher power of (1 − ΛZt,i ci ). Thus, the nonlinear term of V artP [Zt+1 14
dependence in the mean achieved by one’s choice of ΛZt effectively determines the structure of the nonlinearity of the conditional second moments. This specialized structure is the discrete-time counterpart to the similarly special structure of moments implied by diffusion models. An interesting question for future research is the feasibility of working with even richer pricing kernels, while preserving the tractability of the resulting (DAM (N ), Λ) models. Though we have allowed for considerable flexibility in specifying the dependence of Λt on Xt , it is desirable to impose sufficient structure on Λt to ensure that the maximum likelihood estimator of ΘP has a well-behaved large-sample distribution. One property of the P distribution of X that takes us a long ways toward assuring this is geometric ergodicity.20 That X will not be a geometrically ergodic process for all specifications of Λt can be seen immediately from (40). If ΛZt,i approaches c1i as Zti increases, then the second term eventually dominates and the state variable is explosive under P. Such explosive behavior is ruled out by geometric ergodicity since, intuitively, the latter ensures that a Markov process converges to its ergodic distribution at a geometric rate. The following proposition provides sufficient conditions for the geometric ergodicity of an autoregressive gamma process (see Appendix A for the proof). Proposition 2 (G.E.(Z)) Suppose that the market price of risk ΛZ (Zt ) is a continuous function of Zt , and the eigenvalues of the matrix ρ, ψi (i = 1, 2, . . . , M ), satisfy maxi |ψi | < 1. If, in addition, 1. ΛZ (z) ≤ 0 for ∀z ≥ 0, or ¯ ≤ 0 as z → ∞ and ρij = 0 for 0 ≤ i 6= j ≤ M , 2. ΛZ (z) → λ then Zt is geometrically ergodic under both Q and P. Establishing geometric ergodicity for the entire state vector Xt is more challenging, because of the range of possible specifications of ΛY t , many of which lie outside those considered in the literature on geometric ergodicity. For this reason researchers will most likely have to treat the issue of geometric ergodicity on a case-by-case basis, as we do in our illustrations. Finally we note that, for our particular choice of Radon-Nykodym derivative, there is also a computationally fast way to simulate directly from the conditional P distribution of X. Specifically, returning to the exponential-affine representations (4) and (32) for the conditional MGFs, upon making the dependence of the coefficients a(·) and b(·) of φQ on the risk-neutral parameters explicit by writing a(u) = a(u; ΘQ ), b(u) = b(u; ΘQ ), ΘQ = (ci , ρi , νi ; µ0 , µ, h0 , hi : i = 1, 2, . . . , M ) , the coefficients A(u, v) and B(u, v) of φP can be written as A(u, v) = a(u; ΘP (v)), B(u, v) = b(u; ΘP (v)), ΘP (v) = (ci (v), ρi (v), νi ; µ0 (v), µ(v), h0 , hi : i = 1, 2, . . . , M ) . 20
See Duffie and Singleton (1993) for definitions and applications of geometric ergodicity in the context of generalized method of moments estimation. General criteria for the geometric ergodicity of a Markov chain have been obtained by Nummelin and Tuominen (1982) and Tweedie (1982).
15
where v 0 = (vZ0 , vY0 ), for M × 1 vector vZ and (N − M ) × 1 vector vY , and ci ρi , ρi (v) = , 1 − vZ,i ci (1 − vZ,i ci )2 ¡ ¢ µ0 (v) = µ0 + h00 vY , µY (v) = µZY + {h0i vY }i=1,2,...,M µYY . ci (v) =
It follows that the conditional density under P has exactly the same functional form as that under Q, except that the latter is now evaluated at the (possibly time-varying) parameters ΘP (Λt ). Analogously to the continuous-time case, the volatility parameters {νi }M i=1 (for the M M stochastic volatility factors), and h0 and {hi }i=1 (for the N − M conditional Gaussian factors), are not affected by the measure change. It follows that, given Xt , the value of the state at date t + 1 can be simulated exactly using the Q density, with the parameters adjusted to reflect the state dependence induced by the measure change. Now consider the problem of computing the conditional P-expectation of a measurable function g(Xt+τ ), for any τ > 1, by Monte Carlo methods. Such computations can be approached in either of two ways. First, defining the random variable D πt,t+τ
=
¶D τ µ Y dP j=1
we can write
dQ
,
(42)
t+j−1,t+j
£ ¤ D E P [g(Xt+τ )|Xt ] = E Q g(Xt+τ )πt,t+τ |Xt .
(43)
The expectation on the right-hand-side of (43) can be computed, for a given value of Xt , by simulation under Q using the known density f Q (Xt+1 |Xt ). Moreover, the nonlinearity in D the P distribution– its non-affine structure– is captured through the random variable πt,t+τ which is also known in closed form. Alternatively, using the preceding short-cut to simulating from the P distribution of X directly, we can compute the left-hand side of (43) by Monte Carlo simulation without reference to the right-hand side. This second approach is used in our empirical illustrations in Section 8.
7
The P Distribution in the Habit-based DT SM
To complete the specification of our habit-based DT SM , it remains to specify the market prices of risk and derive the physical distribution of bond yields. We take up these issues in this section, along with discussions of steady-state conditions and the continuous-time limit of our discrete-time model. The latter facilitates comparison with the habit-based models studied by CC and Wachter.
16
7.1
The Market Price of Risk in the Habit-based DT SM
Substituting (17) into (3) leads to µ
− mt,t+1
σg = −γ 1 + Q σt [zt+1 ]
¶ zt+1 + πt+1
−log δ + γzt + γf (zt ) + γσg
EtQ [zt+1 ] . σtQ [zt+1 ]
(44)
Since the market price of risk21 Λt is, by definition, the loading on Xt in mt+1 , ³ ´ # " −γ 1 + σQ [zσg ] Λt = . t t+1 1
(45)
It follows that the market price of inflation risk is constant at 1 and the market price of inverse surplus consumption risk is time-varying and (potentially highly) nonlinear in zt . The corresponding physical density of Xt+1 is given by: 0
0
eΛt [zt+1 ,πt+1 ] f (zt+1 , πt+1 |zt , πt ) = f (zt+1 , πt+1 |zt , πt ) × Q . φ (Λt ; [zt , πt ]) P
Q
(46)
In implementing the ML estimator using this physical density, we constrain the parameters of our model to rule out non-stationarity and an absorbing boundary for surplus consumption. The following proposition gives conditions under which the state variables are geometrically ergodic and zt is non-absorbing at zero. Proposition 3 If
√
ρz − 1 − γ, ρπ ∈ (0, 1), and vz ≥ 1, (47) cz then the state variables zt and πt are geometrically ergodic and non-absorbing at zero. σπ,g >
Proof: See Appendix B. It should be noted that Proposition 3 only gives sufficient conditions. Owing to the nonlinear dynamics under the physical measure, we have not discovered a set of necessary conditions for ergodicity. Simulations at various parameters values, however, suggest that the conditions of Proposition 3 are close to being necessary. Even slight violations of these constraints will often result in explosive behavior of the state variables. 21
Consistent with the earlier sections, the concept of the market price of risk used here refers to the price per unit of variance of the state variables.
17
7.2
Steady State Conditions
Following CC and Wachter, we require that: ¯ ¯ ∂ log Ht+1 = 0¯¯ , ∂ct+1 zt =¯ z ¯ ¯ ∂ (∂ log Ht+1 /∂ct+1 ) = 0¯¯ . ∂zt zt =¯ z
(48) (49)
As explained by CC, the first condition guarantees that the (log) habit level log Ht is a deterministic function of past consumption around the steady state (¯ z ). The second condition ensures that this deterministic function is locally increasing in past consumption. As shown in Appendix C, these conditions impose the following constraints on the model parameters: ABρz , 1 + 2Bρz (A − 2¯ z )ρz = , cz = z¯ + log(1 − A),
(50)
z¯ = νz smax
(51) (52)
where A=1+
7.3
s
σg2 2cz ρz
−
σg2 σg4 + 2 2 cz ρz 4cz ρz
¢ ¡ 1 + Aγ + σπ,g cz and B = ¡ . ¡ ¢ ¢2 1 + Aγ + σπ,g cz − ρz
(53)
The Continuous Time Limit
To put the model parameters in connection with the time interval ∆, let: 1 2κz θz ρz = 1 − κz ∆; cz = σz2 ∆; νz = 2 σz2
√ ρπ = 1 − κπ ∆; ρπ,z = −κπ,z ∆; σπ = σπc ∆ √ σg = σgc ∆; C = C c ∆.
(54)
Proposition 4 In the continuous time limit, Xt0 = (zt , πt ) follows the risk-neutral process Q Q dXt = (κθQ X − κX Xt )dt + ΣSz,t dBX,t ,
(55)
where · κθQ X
=
κz θz κπ π ¯ + κπ,z θz
¸
· and
18
κQ X
=
κz 0 κπ,z κπ
¸ ,
· Σ=
σz 0 −σπ,g σz σπc
¸
· √ and Sz,t =
zt 0 0 1
¸ .
Under the P measure, Xt follows the process √ P dXt = (κθPX − κPX Xt − φ zt )dt + ΣSz,t dBX,t ,
(56)
where · κθPX
=
κθQ X
+
0 σπc 2
·
¸ ,
κPX
=
κQ X
+
σz2 (γ
+ σπ,g )
1 0 −σπ,g 0
·
¸ , and φ =
The consumption growth process approaches the diffusion £ √ ¤ Q gt = C c − (γ + σπ,g )σgc σz zt dt − σgc dBz,t
γσgc σz
1 −σπ,g
¸ .
(57)
under Q, and the process P gt = (C c + γσgc 2 )dt − σgc dBz,t
(58)
under the historical distribution. Proof: See Appendix E. Proposition 4 confirms that the states processes under our formulation are exponentially affine under Q. Moreover, from equation (56), it is seen that the nonlinearity in the drift of the physical state processes takes a particularly simple form: it depends on the square-root of the inverse consumption surplus zt . This form of nonlinearity bears close resemblance to that considered by Duarte (2004). Whereas Duarte (2004) studies a reduced-form model with the coefficient on the nonlinear term being a free parameter, our structural setup defines this coefficient in terms of the underlying parameters of the model. For example, the coefficient √ of zt for the P-process of zt is the product of γ, σgc and σz . Since the P-nonlinearity of zt arises from the nonlinear risk premiums implied by habit formation, it is intuitive that the nonlinear component in the P-drift of zt is a function of the utility curvature γ (which modulates the price of risk) and σgc and σz (which modulate the quantities of risks). Note also from equation (58) that consumption growth gt approaches a homoskedastic process with a constant mean under P. This justifies the choice we made earlier in modeling the risk-neutral conditional expectation of gt+1 , f (zt ). Specifically, to align our model to those of CC and Wachter, f (zt ) is chosen so that its nonlinearity is exactly netted out by the nonlinearity generated by the habit-based market prices of risk, giving a homoskedastic P-process with a constant mean. More generally, allowing for some degree of predictability of consumption growth under P is feasible within our modeling framework. For example, with a slight modification to f (zt ), the P-drift of gt+1 could be driven by zt in a manner very similar to the long run risk model of Bansal and Yaron (2004). 19
8
Empirical Illustrations
Previous studies of habit-based models of asset prices have typically focused on parameters chosen by matching model-implied moments to a selected set of sample moments of the data.22 As we document subsequently, the degree to which habit-based models resolve puzzles in the bond pricing literature depend on which of seemingly equally sensible sets of moments are used in calibration. This sensitivity motivates our interest in examining the properties of our model evaluated at the maximum likelihood (ML) estimates of the model. The likelihood function implicitly uses all of the moments of the distributions of the variables in the model, weighted by the precision with which they are estimated. ML estimation is relatively challenging in Wachter (2005)’s formulation of the habit-based model, owing to the nonlinear dependence of bond yields on the state. Within our framework, joint ML estimation of all model parameters is feasible since both analytical bond prices and likelihood function are available. Summarizing the estimation problem, the Q distribution of the inverse surplus consumption ratio zt , a CIR-like process, is governed by three parameters: the persistence parameter ρz , the volatility parameter cz , and risk-netural long-run mean of zt , νz . Whereas ρz and cz are free parameters, νz is determined as function of other parameters of the model (to satisfy the steady-state conditions described in Appendix C). Similarly, ρπ , σπ and θπ govern the persistence, volatility and long run mean of the risk-neutral inflation process. In addition, the contemporaneous correlation and feedback effect between zt and πt are captured through σπg and ρπg , respectively. δ is the subjective discount factor. δ0 is the constant term in the short rate equation. Finally, γ determines the curvature of the habit utility function.
8.1
Data
We follow Piazzesi and Schneider (2007) and construct our quarterly measures of inflation and real consumption from the NIPA price and quantity indexes.23 Compared to the CPI index which covers a wide basket of goods, our inflation measure maps precisely to the measure of aggregate consumption used in the analysis. Only consumption of non-durable goods and services is included. Total real consumption is divided by the corresponding population series, obtained from the Census Bureau. To reduce the level of measurement noise in the inflation series, we follow the suggestion of Kim (2008) and process our inflation series through an ARMA(1,1) filter: πt = (1 − 0.924)0.010 + 0.924πt−1 + ²t − 0.346.²t−1 22
(59)
Wachter (2005), for example, calibrates her model through a two-step process: (1) parameters governing the physical dynamics of consumption growth and inflation are estimated using limited-information M L methods (the constraints imposed by the pricing model are not enforced); (2) given these estimated parameters, other parameters of the model are calibrated to match certain moments of the data. This two-step procedure, also adopted by Boudoukh (1993), reduces the size of the parameter space in calibration, thereby alleviating the computational burden in numerically computing bond prices. 23 NIPA tables 2.3.3, 2.3.4 and 2.3.5.
20
We then use an exponentially smoothed measure of observed inflation: (0.924 − 0.346)
∞ X
0.346j (πt−j − 0.010) + 0.010
(60)
j=0
as the true inflation series purged of measurement noise.24 The interest rate data are downloaded from the Federal Reserve’s web page accompanying Gurkaynak, Sack, and Wright (2006).25 Available maturities are in whole numbers of years, ranging from one to seven years. Our analysis is performed using quarterly data over the sample period 1961 through 2007.
8.2
Calibration
As an informative first-step towards the analysis of our habit-based DT SM we calibrate the model to various sample moments in the data in order to explore the sensitivity of the model’s properties to alternative choices of parameter values. We choose σg to match the standard deviation of consumption growth in the data. Then, for each set of {γ, ρz , cz , ρπ , ρπ,z , σπg , σπ }, we compute νz from the steady state conditions described in Appendix C. Given ρz , cz , νz , we simulate a long series of zt , and choose θπ to match the sample mean of inflation, ET [πt ].26 δ0 is chosen to match the observed level of the yield curve, defined as the midpoint between the mean of the 1-year zero yields and the 7-year zero yields. Next, we choose δ to match the sample mean of consumption growth, ET [gt ].27 Finally, we choose {γ, ρz , cz , ρπ , ρπ,z , σπg , σπ } to match sample moments from the data according to one of the following two schemes. In the first scheme (CS), we place positive weights on the sample means and volatilities of interest rates, inflation volatility, inflation persistence, the unconditional correlation between consumption growth and inflation, and 24
We also re-ran our analysis using the raw inflation series and found no significant qualitative changes. Motivated by similar considerations, Wachter (2005) also uses an ARMA filter to process her inflation and consumption data. 25 http://www.federalreserve.gov/Pubs/feds/2006/200628/200628abs.html 26 The simulation size is 50,000, after a burn-in sample of 5,000. It can be shown that: µ µ ¶ ¶ 1 νz cz θπ = ET [πt ] − ρπz ES [zt ] − − ES [zt ]σπg (1 − ρz ) + (σπ2 + σπg νz cz ) , 1 − ρπ 1 − ρz where ES [.] denotes averaging over simulated values. 27 Precisely, we match ¶ µ 1 2 1 2 2 1 νz cz 2 2 − σπ + γ σg − νz cz (γ + σπg ) . log(δ) = γET [gt ] − δ0 + γνz cz − θπ (1 − ρπ ) + ρπz 1 − ρz 2 2 2 This equation matches the drift of the continuous-time consumption growth process to the discrete-time sample mean. This is convenient since a closed-form expression for average consumption growth is only available in the continuous time limit. As shown in Table 2 the errors from not using the model-implied mean in discrete time are small.
21
the Campbell and Shiller (1987) (CS) regression coefficients.28 In the second scheme (VO), we adopt the same weighing scheme except that we place zero weights on the CS regression coefficients (the slope coefficient in the regression of changes in long-term bond yields on the slope of the yield curve). The calibrated values of the parameters are displayed in Table 1.
γ δ σg ρz cz ρπ θπ ρπz σπg σπ δ0 νz z¯ smax
Calibration Scheme CS Scheme VO 2.1977 5.0000 0.9904 1.0134 0.0044 0.0044 1.0162 1.0012 0.0070 0.0006 0.8941 0.8957 0.0000 -0.0348 0.0015 0.0021 -0.0223 -0.1019 0.0001 0.0005 0.0053 0.0017 1.0819 4.0318 0.471 0.416 -2.51 -1.38
Estimates 2.4005 0.9697 0.0048 1.0273 0.0120 0.9467 0.0131 -0.0001 -0.0042 0.0021 0.0043 1.2869 0.471 -2.69
ML Estimation Asymptotic s.e. Small-sample s.e. 0.1230 0.1369 0.0027 0.0042 0.0002 0.0008 0.0015 0.0038 0.0002 0.0015 0.0065 0.0091 0.0016 0.0019 0.00003 0.00005 0.0023 0.0050 0.0001 0.0007 0.0003 0.0008 -
Table 1: Parameters values from calibration and from full-information ML estimation of the habit-based model. Small-sample standard errors are computed using M L estimates from 100 simulated samples with a length of 185 quarters.
The moments of the consumption growth and inflation process corresponding to two calibrated parameter sets are reported in Table 2. Both calibrated models do a good job of capturing the moments of inflation and consumption growth in the data.29 E[gt ] σ(πt ) corr(gt , πt ) corr(πt , πt+1 )
Data 0.0053 0.0059 -0.3382 0.9324
Scheme CS 0.0054 0.0058 -0.2670 0.9320
Scheme VO 0.0053 0.0062 -0.3392 0.9320
Table 2: Sample and Model-Implied Moments However, notable differences between the models emerge when we examine the modelimplied moments of bond yields. For each calibration scheme, the three graphs in Figure 1 display, from left to right, the sample and model-implied average yield curve, term structure 28
We minimize the sum of squared differences between the model-implied and sample moments. To ensure that the yield curve is matched on average, we multiply the difference in means of interest rates by a factor of 10 before computing the sum of squared errors. All other moments receive a weight of 1. 29 The volatility of consumption growth and the long run mean of inflation are not reported since they are perfectly matched as part of our calibration process.
22
Yield curve
Volatility curve
0.075
Campbell Shiller Regression
0.035 Data Model
0.07
0 −0.5
0.03
−1 0.065
0.025 −1.5
0.06
0.055
0.02
0
2 4 6 Maturity (years)
8
0.015
−2
0
2 4 6 Maturity (years)
8
−2.5
2
4 6 Maturity (years)
8
(a) Scheme CS Yield curve
Volatility curve
0.075
Campbell Shiller Regression
0.035
2
0.07
0.03
1
0.065
0.025
0.06
0.02
0.055
0.015
Data Model
0 −1 −2
0
2 4 6 Maturity (years)
8
0
2 4 6 Maturity (years)
8
2
4 6 Maturity (years)
8
(b) Scheme VO
Figure 1: Moment Matching from Calibration Schemes: from left to right, average yields, volatility of yields, and Campbell-Shiller regression coefficients.
of volatility, and CS regression coefficients. Focusing first on scheme CS (Figure 1(a)) the model-implied CS regression coefficients match strikingly well with those in our sample. This near perfect match does not come without compromising the fit to other moments. In particular, scheme CS produces an upward sloping volatility curve which is contrary to what is seen in the data: yield volatilities decay with maturity. Under scheme VO (Figure 1(b)), the sample average yield curve and volatility curve are matched perfectly. However now, the model completely fails to match the CS coefficients– as if the expectations theory holds in this model. In the light of the sensitivity of fit to the choice of calibration scheme documented in Figure 1, we turn next to an exploration of the fit based on full-information ML estimates of the parameters.
23
8.3
M L Estimation
For each quarter in our sample, we compute the inverse consumption surplus ratio, zt+1 , from equation (17), based on our observation of gt+1 and the previously implied value of zt : zt+1 = EtQ [zt+1 ] −
σtQ [zt+1 ] (gt+1 − f (zt )). σg
(61)
The physical density f P (zt+1 πt+1 |zt , πt ) is then computed using equation (46). In addition, we assume that bonds with one, four and seven years to maturity are priced with normally distributed i.i.d. errors with mean zero and constant variances. This distributional assumption for the pricing errors introduces minimal additional flexibility in fitting yields, beyond that inherent in the habit-based DT SM . Combining these observations, and letting Rt denote the continuously compounded yields on these three bonds, the likelihood of the observed time series {gt , πt , Rt0 } is L({gt , πt , Rt }Tt=2 )
=
T Y σtQ [zt+1 ] t=2
σg
1 4 7 f P (zt+1 , πt+1 |zt , πt )f P (Rt+1 , Rt+1 , Rt+1 |zt+1 , πt+1 ),
(62)
where the first term on the right-hand side of (62) is the Jacobian of the transformation between gt and zt .30 The resulting estimates and their associated standard errors are reported in the last three columns of Table 1. All of the parameters are estimated with considerable precision. The point estimate of the utility curvature parameter (γ) is 2.4 - a value quite close to what is adopted in studies of the equity premium and Wachter’s choice of 2. Likewise, the steady-state value of zt (¯ z) and the upper boundary of st (smax ) are very close in magnitude to those used by CC and Wachter. Those parameters associated with risk-neutral distribution of the state are not directly comparable to the values in previous studies. Moreover, the model-implied fitted values of surplus consumption and habit both seem plausible. From Figure 2(a) it is seen that st co-moves strongly with the business cycle, with four noticeable troughs corresponding to recessions in 1975, 1982, 1991 and 2002. The time-series behavior of Ht (Figure 2(b)) is very much in line with our expectations: it is smooth, persistent and increasing with the level of consumption. Having established that our model fits many features of the macro variables well, we turn next to an exploration of the fit to moments on bond yields. Figure 3 displays modelimplied population term structures of the means and volatilities of bond yields (“Long run”), 30
In maximizing this likelihood function we address the possibility of negative fitted values of zt by assuming that any such negative values of zt are the manifestation of an exponentially distributed error. In this manner errors in zt that lead to negative fitted values are continuously penalized and, in the presence of such errors, the likelihood function remains smooth. In our sample there were a small number of negative fitted z’s, all of which occurred prior to 1974. Also, to ensure that our estimates are global optima we implement the optimization in two steps. First, we randomly generate thousands of starting points, quickly improve them within a short time window and then rank them in the order of likelihood value. Second, we use the best 500 parameter sets as starting points and numerically maximize the likelihood function (62) until convergence. Out of these 500 local optima we select the parameter set that yields the highest likelihood value.
24
Time series of z
Time series of Consumption and Consumption Habit
t
250
1
Consumption Level Habit Level
0.8 0.6 200
0.4 0.2 0 1960
1970
1980
1990 Time Time series of st
2000
2010 150
−2.5 −3
100
−3.5 −4 1960
1970
1980
1990
2000
50 1960
2010
Time
1970
1980
1990
2000
2010
Time
(a) Consumption Surplus Ratio
(b) Consumption Habit
Figure 2: Time Series of Consumption Surplus Ratio (st ) and Consumption Habit (Ht ) their sample counterparts (“Data”), and fifth and ninety-fifth percentiles of the small-sample distributions of these statistics. The latter are computed by simulating 5000 sample paths of length 185 quarters (the size of our sample of bond yields) and, for each sample path, computing the moments of bond yields. The means of the small-sample distributions of these moments are very similar to their population counterparts, and so they are omitted to avoid congestion in these figures. The level and the slope of the yield curve, as well as the term structure of volatilities of yields, are reasonably well matched. Although the long end of the population mean yield curve is higher than its sample counterpart, the latter is bracketed by the the 5th and 95th percentiles of the small-sample distribution of the sample means. The population term structure of volatility (Figure 3(b)) lies below our sample counterpart, perhaps owing in part to the fact that bond yields are priced with error in our setup. Nevertheless, the model clearly captures the pronounced downward slope in the volatility curve. Additionally, the percentiles of the model-implied small-sample distribution of volatilities come close to bracketing the sample estimates, even without adding on the volatilities of the pricing errors. We are less successful at replicating the failure of the expectations hypothesis. From Figure 4 it can be seen that the population CS regression coefficients (“Long-run mean”) lie below one and exhibit a decreasing pattern. However, this line is substantially above the historically estimated coefficients (marked “Data”), and even the 5th percentile values of the small-sample distribution lie well above the sample coefficients. Why does the habit-based DT SM fail to resolve the expectations puzzle? Letting ξtn denote the expected excess return from holding the n-period bond for 1 period, the CS regression coefficients are cov(Rtn − Rt1 , ξtn ) . (63) φn = 1 − n var(Rtn − Rt1 ) 25
Yield curve 0.09
0.035 Data Long run volatility
0.085
5th percentile 0.03
0.08
95th percentile
0.075 0.025 0.07 0.065 0.02 0.06 Data Long run mean
0.055
0.015
5th percentile
0.05
95th percentile 0.045
1
2
3
4 Maturity (years)
5
6
0.01
7
(a) Mean Yield Curves
1
2
3
4 Maturity (years)
5
6
7
(b) Term Structure of Volatility
Figure 3: Sample, population, and small sample distributions of means and volatilities of bond yields. 4
Data Long run mean
3
5th percentile 2
95th percentile
1
0
−1
−2
2
3
4 5 Maturity (years)
6
7
Figure 4: Campbell-Shiller Regressions To generate negative φn as required by the data a model needs to produce a positive correlation between the slope of the yield curve and ξtn . Now for n-period bond with the corresponding zero yield RTn = an + bn zt + cn πt , its one-period expected excess return is approximately (see Section 6): ³ ´ # ¸" · σg 2 2 £ ¤ √ −σπg σz zt σz zt −γ 1 + σz zt −bn −cn (64) ξtn ≈ 2 σz2 zt + σπ2 −σπg σz2 zt σπg 1 £ √ ¤ (65) = (bn − cn σπg ) (γ + σπg )σz2 zt + γσg σz zt − cn σp2 , where σz =
√
2cz . For the yield curve to be upward sloping on average, ξtn must be positive
26
an
bn
cn
0.04 ML Scheme CS
3.5
0.035 3 0.03 2.5
0.025 0.02
2
0.015 1.5 0.01 2
4 6 Maturity (years)
8
0.005
0
2
4 6 Maturity (years)
8
1
0
2
4 6 Maturity (years)
8
Figure 5: Loadings: Rtn = an + bn zt + cn πt on average, which requires (bn − cn σπg ) > 0.31 This, in turn, makes ξtn an increasing function of zt . As a result, for the slope of the yield curve to be positively correlated with ξtn , it must be positively correlated with zt . Ignoring the constant term, the slope of the yield curve is (bn − b1 )zt + (cn − c1 )πt . For the first term to contribute positively to corr(ξtn , Rtn − Rt1 ), bn − b1 must be positive and this, in turn, calls for a risk-neutral mean reversion parameter of zt (ρz ) greater than 1. From Table 1, ρz is calibrated at 1.0162 under the CS scheme and estimated at 1.0273, thus both generating an increasing pattern of bn (see Figure 5). Turning to the second term, since cn − c1 < 0, it will contribute positively to resolving the expectations puzzle if corr(zt , πt ) < 0. However, corr(πt , gt ) < 0 and zt is conditionally perfectly negatively correlated with gt , so inducing cov(zt , πt ) < 0 within this habit model would be quite challenging. From simulations, corr(zt , πt ) is indeed positive (0.0024) under the CS scheme. With the M L estimates, corr(zt , πt ) is just negative (-0.0001). Comparing the calibrated parameters for Scheme CS to corresponding the M L estimates, it is striking how closely many of them match up. Yet, as we have seen, these two parameter sets have very different implications for the moments of bond yields. Key to understanding this difference is by comparing the patterns of loadings across the different parameter sets. The fact that corr(zt , πt ) > 0 under the CS scheme means that bn has to increase quite fast in n to, first, offset the negative correlation generated from (cn − c1 )πt and, second, create a positive correlation between ξtn and zt . However, this very effort to generate an increasing pattern in bn contributes to an increasing pattern in bond yield volatility var(Rtn ) = b2n var(zt ) + c2n var(πt ) + 2bn cn cov(zt , πt ). Omitting the first (increasing) component (b2n var(zt )), we confirm through simulations that the other two components (c2n var(πt ) + 2bn cn cov(zt , πt )) decrease in maturity n under the CS scheme. Likewise, in order to match the downward sloping pattern of volatilities, the M L estimates generate a slowly increasing pattern of bn . However, this proves to be too slow to induce sufficient positive correlation between ξtn and zt to match the data. To see this graphically we plot the implied bn − b1 and bbn1 for the two calibration schemes and the M L estimates in Figure 6. The steep slopes of bn − b1 as well as bbn1 distinctly set the CS scheme apart from 31
Strictly speaking, to have (bn − cn σπg ) > 0, we also need γ + σπg > 0. This is guaranteed, from the ergodicity condition and the requirement that ρz > 1 discussed in the text.
27
b −b n
b /b
1
n
0.035
1
6 ML Scheme CS Scheme VO
0.03
5
0.025 4
0.02 0.015
3
0.01 2 0.005 0
0
2
4 6 Maturity (years)
1
8
0
2
4 6 Maturity (years)
8
Figure 6: Comparison of bn the V O scheme and the M L estimates. This pattern of the loadings bn helps the CS calibration scheme match the CS regression coefficients better, but at the expense of not matching the volatility curve. From Table 4 in Wachter (2005) it is seen that, at her calibrated value of the parameters, her habit-based model also produces a counter-factual upward sloping term structure of volatility. While we note that there are differences between our setup and that of Wachter (2005), given the similarity in how risk-premiums are generated under the two economies, it is quite possible that the same tension that we find here extends to Wachter (2005)’s setup as well. Though the literature on evaluating the ability of equilibrium DT SM s to resolve the expectations puzzle has focused largely on the CS coefficients, Dai and Singleton (2002) show that a successful model should also replicate the risk-premium adjusted regressions coefficients. That is, in the regressions n−1 Rt+1 − Rtn +
1 Rn − Rt1 n Et [ξt+1 ] = constant + φn t + residual, n−1 n−1
(66)
the coefficients φn should be one for all maturities. Figure 7 displays these adjusted coefficients for model-implied yields evaluated at our ML estimates as well as at the calibrated parameter values. The ML premium-adjusted coefficients come closest to having φn = 1, but all of the estimates fall well below their theoretical value. In summary, though our illustrative habit-based DT SM produces time varying expected excess returns, the magnitude and nature of this predictability is not sufficient to resolve the expectations puzzles in the literature. Central to this failure is the inherent tension between matching the slope of the volatility curve and the time series behavior of expected excess returns.32 There is room for improvement. In its current form, the surplus consumption ratio is the lone driver of expected excess return. As such, the distribution of zt is largely responsible for matching both the the volatility structure, the slope of the yield curve as well 32
This finding is reminiscent of the result in Dai and Singleton (2002) that an A1 (3) reduced-form DT SM was unable to resolve the expectations puzzles.
28
0.5 Data ML Scheme CS Scheme VO1 Scheme VO2
0
−0.5
−1
−1.5
−2
2
3
4 5 Maturity (years)
6
7
Figure 7: Risk-Premium Adjusted Regression Coefficients as the CS regression coefficients. Meanwhile, inflation contributes very little to the overall risk-premium dynamics since the price of inflation risk is constant at 1 and the volatility of inflation-specific shocks is constant. Introducing more flexibility along either or both of these dimensions would likely assist the habit-based model in matching the distribution of bond yields.33
9
Concluding Remarks
In this paper we have argued that, along important dimensions, researchers can gain flexibility and tractability in analyzing DTSMs by switching from continuous to discrete time. We have developed a family of nonlinear DTSMs that has several key properties: (i) under Q, the risk factors X follow the discrete-time counterpart of an affine process residing in one of the families AQ M (N ), as classified by Dai and Singleton (2000), (ii) the pricing kernel is specified so as to give the modeler nearly complete flexibility in specifying the market price of risk Λt of the risk factors, and (iii) for any admissible specification of Λt , the likelihood function of the bond yield data is known in closed form. This modeling framework was illustrated by estimating a nonlinear (non-affine under P), equilibrium DT SM in which agents’ preferences exhibit habit formation. A novel feature of our formulation is that we posit an affineQ representation of the state Xt , and choose the consumption process under the historical measure so that the one-period bond yield is an affine function of Xt . As such, an equilibrium implication of our model is that bond yields are 33
Gallmeyer, Hollifield, Palomino, and Zin (2007), for example, consider an inflation process that is endogenously determined to satisfy the Taylor rule.
29
known in closed form, even though preferences are nonlinear and the state exhibits stochastic volatility. The market prices of risk associated with our habit-based preferences imply that the surplus consumption ratio follows a nonlinear (non-affine) process under the historical measure. Nevertheless, the likelihood function of the data is known in closed form. The tractability of likelihood-based estimation means that our approach offers an attractive alternative estimation strategy to the calibration methods most often applied in the study of equilibrium asset pricing models. As is illustrated in our empirical analysis, calibration can easily lead to parameters that render models equally effective at matching salient features of the macroeconomic series while having fundamentally different implications for asset prices. Focus on the likelihood function provides one, systematic means of incorporating full information about the conditional joint distribution of the macroeconomic variables and asset returns. Our framework is applicable more generally to other equilibrium DT SM s and also offers a means of exploring richer no-arbitrage, reduced form models.34 Key to this applicability is the presumption that the state variables follow an affine process under Q. Many of the current generation of macro-finance models of the term structure either presume an affineQ state process or they are easily reformulated to have this structure (seemingly) without changing their essential properties. We note in particular that this assumption is explicit in many of the macro-finance models of the term structure being developed at central banks (e.g., Rudebusch and Wu (2008) and Hordahl, Tristani, and Vestin (2006)), as well as in models with long-run risks based on the framework in Bansal and Yaron (2004). Our framework provides a means of enrichening the data-generating processes in these and related studies.
34
Furthermore, under certain conditions analogous to those set forth in Dai and Singleton (2003) for continuous-time models, we preserve analytical bond pricing even in the presence of switching regimes. Ang and Bekaert (2005) and Dai, Singleton, and Yang (2005) study DTSMs in which X follows a regime-switching DAQ 0 (N ) process, with the latter study allowing for priced regime-shift risk. Monfort and Pegoraro (2006) propose several families of regime-switching, affine models based on Gaussian and autoregressive gamma models.
30
Appendix
A
Proof of Proposition G.E.(Z)
The proof follows from a lemma due to Mokkadem (1985) Lemma 1 (Mokkadem) Suppose {Zt } is an aperiodic and irreducible Markov chain defined by Zt+1 = H(Zt , ²t+1 , θ), (67) where ²t is an i.i.d. process. Fix θ and suppose there are constants K > 0, δθ ∈ (0, 1), and q > 0 such that H(·, ²1 , θ) is well defined and continuous with kH(z, ²1 , θ)kq < δθ kzk,
kzk > K.
(68)
Then {Zt } is geometrically ergodic. In our setting, we can write, without loss of generality, £ ¤ p H(z, ²1 , θ) = a(1) (λ(z)) + b(1) (λ(z))z + Ω(z)²1 , where ²1 has a zero mean and unit variance, and Ω(z) = a(2) (λ(z)) + b(2) (λ(z))z. Take q = 2, we have p kH(z, ²1 , θ)k2 ka(1) (λ(z))k kb(1) (λ(z))zk k Ω(z)²1 k2 ≤ + + . (69) kzk kzk kzk kzk The first term on the right-hand-side of (69) satisfies h i νi ci kvec k (1) 1−λi (z)ci ka (λ(z))k kvec [νi ci ] k = ≤ → 0, kzk → ∞, kzk kzk kzk
(70)
where we have used the assumption (i) to obtain the inequality. Since all elements of ρ are non-negative, if 1 − λi (z)ci ≥ 1 for all z and i, then the second term in (69) is bounded by kb(1) (λ(z))zk kρzk ≤ ≤ max |ψi |. i kzk kzk If, in addition, ρij = 0 for i 6= j, the above bound is valid for each element of z when it is sufficiently large. That is, there exists a K > 0, such that kρii zi k kb(1) ii (λ(z))zi k ≤ ≤ ρii ≤ max ψi , zi > K i kzi k kzi k
31
Finally, the last term in (69)pcan be made arbitrarily small by choice of a √ sufficiently large K, because k²1 k2 = 1 and Ω(z) depends on z through terms of the form z.35 The only term on the right-hand side of (69) that does not become arbitrarily small as K increases towards infinity is the second term. Since we assume that maxi |ψi | < 1, we are free to choose δθ to satisfy maxi |ψi | < δθ < 1 so that Lemma 1 is satisfied.
B
Proof of Proposition 3
From equation (46), we have: 0
0
eΛt [zt+1 ,πt+1 ] φQ (Λt ; [zt , πt ]) eΛz,t zt+1 E Q [eπt+1 |zt+1 , zt , πt ] = f Q (zt+1 |zt ) × φQ (Λt ; [zt , πt ]) eπt+1 ×f Q (πt+1 |zt+1 , zt , πt ) × Q πt+1 E [e |zt+1 , zt , πt ] e(Λz,t −σπ,g )zt+1 = f Q (zt+1 |zt ) × Q (Λz,t −σπ,g )zt+1 E [e |zt ] eπt+1 Q ×f (πt+1 |zt+1 , zt , πt ) × Q πt+1 E [e |zt+1 , zt , πt ]
f P (zt+1 , πt+1 |zt , πt ) = f Q (zt+1 , πt+1 |zt , πt ) ×
(71)
As such, we have: e(Λz,t −σπ,g )zt+1 E Q [e(Λz,t −σπ,g )zt+1 |zt ] eπt+1 f P (πt+1 |zt+1 , zt , πt ) = f Q (πt+1 |zt+1 , zt , πt ) × Q πt+1 E [e |zt+1 , zt , πt ] f P (zt+1 |zt ) = f Q (zt+1 |zt ) ×
B.1
(72) (73)
Regularity of zt
From equation (72), zt+1 follows an autonomous process under P with an adjusted market prices of risk of Λz,t − σπ,g . For this density to be well-defined, we need to make sure that:36
Substitute Λz,t
1 − (Λz,t − σπ,g )cz > 0 f or all zt > 0. ´ ³ = −γ 1 + σQ [zσg ] , we have: t
(74)
t+1
µ µ 1+ γ 1+
σg Q σt [zt+1 ]
¶
¶ + σπ,g cz > 0, f or all zt > 0,
35
(75)
See Duffie and Singleton (1993) for a discussion of the geometric ergodicity of models in which volatility depends on terms of the form xγ , for γ < 1. By using L2 norm (q = 2), we can apply Mokkadem’s lemma without the i.i.d. assumption for the state innovations. 36 This expression goes under the logarithm operator in the density.
32
which requires 1 + (γ + σπ,g )cz > 0
(76)
Applying (40) and (41), we can write down the first two moments of zt as follows: vz c z ρz + zt 1 − (Λz,t − σπ,g )cz (1 − (Λz,t − σπ,g )cz )2 vz c2z 2cz ρz = + zt 2 (1 − (Λz,t − σπ,g )cz ) (1 − (Λz,t − σπ,g )cz )3
E P [zt+1 |zt ] = σ P [zt+1 |zt ]2
(77)
zt would be geometrically ergodic, according to Proposition G.E.(Z), if we have the limit of: ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ρz vz cz ¯ 1−(Λz,t −σπ,g )cz ¯ + ¯ (1−(Λz,t −σπ,g )cz )2 ¯ |z| + σ P [zt+1 |zt ] (78) |z| strictly less than 1 as z → ∞. 1 is bounded from condition (76), the first and third terms go to zero Since |1−(Λz,t −σ π,g )cz | in the limit. Condition (78) therefore reduces to ¯ ¯ ¯ ¯ ρ z ¯<1 Limzt →∞ ¯¯ (79) (1 − (Λz,t − σπ,g )cz )2 ¯ which, in turn, is equivalent to 1 + (γ + σπ,g )cz >
√
ρz ,
an even stronger condition than (76). With little modification, we have √ ρz − 1 σπ,g > − γ, cz
(80)
(81)
which is precisely the first equation in Proposition 3. In addition, to prevent zt from being absorbed at zero, under Q, a standard requirement is that νz ≥ 1.37 Intuitively, νz controls the relative strength of the mean reverting drift that pulls zt away from its zero boundary and the diffusive force that could possibly absorb zt at zero. To have non-absorbing behavior under Q, the former has to be stronger than the latter, which requires νz ≥ 1. Since P and Q are equivalent measures, νz ≥ 1 also guarantees that zt is non-absorbing under P. Another way of seeing why this is the case is by noting that νz are the same under both P and Q as discussed in section 6. Therefore νz modulates the relative strength of the mean reversion and diffusion forces under both measures. 37
This requirement is very similar in a continuous time setup. For a continuous CIR process characterized by the reversion parameter κ, the long run mean parameter θ and the volatility parameter σ to be nonabsorbing at zero, the usual constraint is 2κθ σ 2 ≥ 1.
33
B.2
Regularity of πt
From equation (73), it follows that πt+1 is Gaussian, conditional on zt+1 , zt and πt with the first two moments given by: EtP [πt+1 |zt+1 , zt , πt ] = π ¯ + ρπ (πt − π ¯ ) + ρπ,z (zt − E Q [zt ]) − σπ,g (zt+1 − EtQ [zt+1 ]) + σπ2 σtP [πt+1 |zt+1 , zt , πt ] = σπ (82) which implies that the auto-regressive coefficients (ρπ ) are the same under both P and Q. If zt is ergodic, therefore, all we need is 0 < ρπ < 1.
C C.1
Derivation of Steady State Conditions ∂xt+1 ∂ct+1
¯ ¯ = 0¯
zt =¯ z
From the definition of surplus consumption ratio, the following identity must hold:
It follows that
Since
∂zt+1 ∂ct+1
=−
xt+1 = ct+1 + log(1 − est+1 ).
(83)
∂xt+1 ∂st+1 /∂ct+1 ∂zt+1 /∂ct+1 =1+ =1− . −s t+1 ∂ct+1 1−e 1 − ezt+1 −smax
(84)
σtQ [zt+1 ] , σg
we have: ∂xt+1 σtQ [zt+1 ] = 1+ ∂ct+1 σg (1 − ezt+1 −smax ) ≈ 1+
σtQ [zt+1 ] . σg (1 − ezt −smax )
(85)
The approximate relation arises since we exploit the fact that zt+1 ≈ zt around the steady state. ¯ ¯ t+1 = 0 In order to have ∂x , therefore, we need: ¯ ∂ct+1 zt =¯ z
σtQ [zt+1 |zt = z¯] = σg (ez¯−smax − 1).
C.2
∂(∂xt+1 /∂ct+1 ) ∂zt
¯ ¯ = 0¯
(86)
zt =¯ z
First, taking the first order derivative of both sides of equation (85), we have: ¡ ¢ ∂σtQ [zt+1 ] ∂ (∂xt+1 /∂ct+1 ) = × σg (1 − ezt −smax ) + σtQ [zt+1 ] σg ezt −smax . ∂zt ∂zt 34
(87)
Substituting (86) into the above equation, evaluated at the steady state value z¯, it follows ¯ ¯ ∂(∂xt+1 /∂ct+1 ) that in order to have = 0¯ we need: ∂zt zt =¯ z
¯
∂σtQ [zt+1 ] ¯¯ ¯ ¯
∂zt
C.3
= σg ez¯−smax .
(88)
zt =¯ z
Steady State Conditions
Together, equations (86) and (88) impose the following constraints: p νz c2z + 2cz ρz z¯ = σg (ez¯−smax − 1) cz ρz p = σg ez¯−smax . 2 νz cz + 2cz ρz z¯ Denoting A =
νz cz ρz
(89)
+ 2¯ z , it can be shown that the above system is equivalent to: σg2 A = 1+ − 2cz ρz
s σg2 σg4 + 2 2 cz ρz 4cz ρz
smax = z¯ + log(1 − A).
(90) (91)
Finally, we define the steady state value of zt as a value z¯ such that: EtP [zt+1 |zt = z¯] = z¯. Since E P [zt+1 |zt ] =
vz cz 1−(Λz,t −σπ,g )cz
+
ρz z, (1−(Λz,t −σπ,g )cz )2 t
(92)
we have:
vz c z ρz + z¯ = z¯, 1 − (Λz¯ − σπ,g )cz (1 − (Λz¯ − σπ,g )cz )2
(93)
where µ
Λz¯
From A =
νz cz ρz
¶¯ ¯ σg ¯ = −γ 1 + Q σt [zt+1 ] ¯zt =¯z µ ¶ 1 = −γ 1 + z¯−smax e −1 γ γ =− . = − νz cz + 2¯ z A ρz
(94) (95) (96)
+ 2¯ z , we have: νz =
(A − 2¯ z )ρz . cz 35
(97)
Substitute νz in (93) and solve for z¯, we have z¯ = where
D
ABρz , 1 + 2Bρz
(98)
¡ ¢ 1 + Aγ + σπ,g cz B=¡ . ¡ ¢ ¢2 1 + Aγ + σπ,g cz − ρz
(99)
Linearity of the Nominal Short Rate
First, according to the Euler equation, the nominal (unannualized) interest rate per unit of time interval is the rt such that: e−rt = EtP [emt,t+1 ] = EtQ [e−mt,t+1 ]−1
(100)
The second part of the identity follows from our construction of the risk-neutral densities and how they connect to their physical counter-parts through the market prices of risks. We have: ¡ ¢ rt = log EtQ [e−mt,t+1 ] . (101) From equations (18) and (44), we can rewrite the pricing kernel as follows: µ ¶ νz cz − mt,t+1 = −log δ + π ¯ + ρπ (πt − π ¯ ) + γzt + ρπ,z zt − − γEtQ [zt+1 ] + σπ ²Q π,t+1 1 − ρz +γf (zt ) + uΛ zt+1 − uΛ EtQ [zt+1 ], where
µ uΛ = −γ 1 +
σg σtQ [zt+1 ]
(102) ¶ − σπ,g .
(103)
Consequently, rt
µ ¶ νz cz 1 = −log δ + π ¯ + ρπ (πt − π ¯ ) + γzt + ρπ,z zt − − γEtQ [zt+1 ] + σπ2 1 − ρz 2 +γf (zt ) + log(EtQ [euΛ zt+1 ]) − uΛ EtQ [zt+1 ]
If 1 f (zt ) = C − (γ + σπ,g )σg σtQ [zt+1 ] − log γ
(104) Ã
EtQ [euΛ zt+1 ] G
EtQ [euΛ zt+1 ]
! ,
(105)
then 1 γf (zt ) = γC − γ(γ + σπ,g )σg σtQ [zt+1 ] − log(EtQ [euΛ zt+1 ]) + uΛ EtQ [zt+1 ] + u2Λ σtQ [zt+1 ]2 . (106) 2 36
Therefore: rt
¶ µ νz cz 1 = −log δ + π ¯ + ρπ (πt − π ¯ ) + γzt + ρπ,z zt − − γEtQ [zt+1 ] + σπ2 1 − ρz 2 1 +γC − γ(γ + σπ,g )σg σtQ [zt+1 ] + u2Λ σtQ [zt+1 ]2 . (107) 2
The expression
µ
σg uΛ = −γ 1 + Q σt [zt+1 ] implies that
¶ − σπ,g
(108)
uΛ σtQ [zt+1 ] = −(γ + σπ,g )σtQ [zt+1 ] − γσg ,
(109)
1 2 Q 1 1 uΛ σt [zt+1 ]2 = (γ + σπ,g )2 σtQ [zt+1 ]2 + γ 2 σg2 + γ(γ + σπ,g )σg σtQ [zt+1 ]. 2 2 2
(110)
so we have
Therefore rt
¶ µ 1 νz cz = −log δ + π ¯ + ρπ (πt − π ¯ ) + γzt + ρπ,z zt − − γEtQ [zt+1 ] + σπ2 1 − ρz 2 1 1 (111) +γC + γ 2 σg2 + (γ + σπ,g )2 σtQ [zt+1 ]2 . 2 2
Since the risk neutral mean and variance of zt+1 are linear in zt , EtQ [zt+1 ] = ρz zt + vz cz σtQ [zt+1 ]2 = 2ρz cz zt + vz c2z ,
(112) (113)
it follows that the short rate is linear in the state variables: rt = δ0 + δz zt + δπ πt .
(114)
Collecting terms, we have: νz cz − γνz cz + γC 1 − ρz 1 1 1 + γ 2 σg2 + (γ + σπ,g )2 νz c2z + σπ2 2 2 2 2 = γ(1 − ρz ) + ρπ,z + (γ + σπ,z ) ρz cz = ρπ .
δ0 = −log δ + (1 − ρπ )¯ π − ρπ,z
δz δπ
37
(115) (116) (117)
E
The Continuous Time Limit
E.1
zt
From our discussion of the DAQ N (N ) in section 3.2, zt follows a CIR process in the time limit under the nominal risk-neutral measure Q: √ Q dzt = κz (θz − zt )dt + σz zt dBz,t . (118) In the limit, ³ the ctotal ´ exposure of the nominal pricing kernel to changes in zt+1 apσg proaches −γ 1 + σz √zt − σπ,g . The first term is the market price of z-risk. The second term accounts for the contemporaneous correlation between zt+1 and πt+1 . This means the difference between the drifts of zt under P and Q must be: µ ¶ γσgc Q P 2 µz,t − µz,t = σz zt −γ − σπ,g − √ . (119) σz zt Under P, It follows that zt approaches the process ¡ √ ¢ √ P dzt = κz θz − (κz + σz2 (γ + σπ,g ))zt − γσgc σz zt dt + σz zt dBz,t .
E.2
(120)
πt
πt will approach the following dynamics under the Q measure: √ Q Q dπt = (κπ π ¯ + κπ,z θz − κπ πt − κπ,z zt ) dt − σπ,g σz zt dBz,t + σπc dBπ,t .
(121)
Since the market price of inflation risk is 1, it follows that Q P dBπ,t = dBπ,t + σπc dt.
(122)
In addition, from the previous section, we know: √ Q P dBz,t = dBz,t − (σz (γ + σπ,g ) zt + γσgc )dt.
(123)
Therefore £ √ ¤ dπt = κπ π ¯ + κπ,z θz + σπc 2 − κπ πt − (κπ,z − σz2 (γ + σπ,g )σπ,g )zt + γσπ,g σgc σz zt dt √ P P . (124) + σπc dBπ,t −σπ,g σz zt dBz,t
E.3
gt
Recalling that f (zt ) = C − (γ +
σπ,g )σg σtQ [zt+1 ] 38
1 − log γ
Ã
EtQ [euΛ zt+1 ] G
EtQ [euΛ zt+1 ]
! ,
(125)
if C = C c ∆, then in the continuous time limit f (zt ) approaches: £ √ ¤ f (zt ) = C c − (γ + σπ,g )σgc σz zt dt.
(126)
Note that the third term of equation (125) disappears in the limit because the two measures Q and QG , by construction, give rise to the same mean and variance - the two moments that matter in a continuous time setup. As a result, in the limit:
Again, applying
gt = dlnCt £ √ ¤ Q = C c − (γ + σπ,g )σgc σz zt dt − σgc dBz,t .
(127) (128)
√ Q P dBz,t − (σz (γ + σπ,g ) zt + γσgc )dt, = dBz,t
(129)
P gt = (C c + γσgc 2 )dt − σgc dBz,t .
(130)
we have
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